Some Computations in the Cyclic Permutations of Completely Rational Nets

Xu, Feng

doi:10.1007/s00220-006-0042-0

Cited by 9 publications

(8 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The final ingredient in the proof of Proposition 1.1 is the following result due to Ng and Schauenburg for modular tensor categories which are centers of spherical fusion categories ( [44]), to Peng Xu for conformal field theories derived from vertex operator algebras (see [63]) and to Coste, Gannon and Bantay for RCFT (see [9,3]). Recall that a congruence subgroup of SL(2, Z) is the kernel of one of the reducing mod m homomorphism SL(2, Z) → SL(2, Z/mZ), for some non-zero integer m.…”

Section: Preliminaries About Modular Tensor Categoriesmentioning

confidence: 99%

See 1 more Smart Citation

Torus bundles not distinguished by TQFT invariants

Funar

2013

Geom. Topol.

View full text Add to dashboard Cite

Abstract. We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in SL(2, Z) and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and lastly by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. On the other side we prove that two torus bundles over the circle with the same quantum invariants are (strongly) commensurable. The examples above show that this is the best that it could be expected.

show abstract

Section: Preliminaries About Modular Tensor Categoriesmentioning

confidence: 99%

“…Find the smallest even integer l ≥ 1 such that Q l = 1. Then (G l−1 , H l−1 ) is the minimal non-trivial solution (u 0 , y 0 ) to the Pell equation (63).…”

Section: Proof the Conjugacy Conditionmentioning

confidence: 99%

Torus bundles not distinguished by TQFT invariants

Funar

2013

Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…Note that any finite group G is embedded in a finite symmetric group S n , and using the theory of permutation orbifolds as in [38] we can always find a completely rational net B such that G acts properly on B and with fixed point subnet A. In this case the intermediate subnets between A and B are in one to one correspondence with subgroups of G. So in this orbifold case the minimal version of Conjecture 1.2 is equivalent to Wall's conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…Note that any finite group G is embedded in a finite symmetric group S n , and using the theory of permutation orbifolds as in [38] we can always find a completely rational net B such that G acts properly on B and with fixed point subnet A. [34] and maximality of conformal inclusions to show that the intermediate subnet is in fact the largest net.…”

Section: Introductionmentioning

confidence: 99%

On Examples of Intermediate Subfactors from Conformal Field Theory

2012

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

Motivated by our subfactor generalization of Wall's conjecture, in this paper we determine all intermediate subfactors for conformal subnets corresponding to four infinite series of conformal inclusions, and as a consequence we verify that these series of subfactors verify our conjecture. Our results can be stated in the framework of Vertex Operator Algebras. We also verify our conjecture for Jones-Wassermann subfactors from representations of Loop groups extending our earlier results.

show abstract

“…The conjecture was later established by Bantay in [Ban03] under certain assumptions. More recently, Xu also solved the conjecture for the modular representation associated to a local conformal net [Xu06].…”

Section: Introductionmentioning

confidence: 99%

Congruence Subgroups and Generalized Frobenius-Schur Indicators

Schauenburg

2010

Commun. Math. Phys.

View full text Add to dashboard Cite

We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay's second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert.Comment: 42 pages Latex, corrected typos, added some references, slightly rewritten abstract of the previous versio

show abstract

Some Computations in the Cyclic Permutations of Completely Rational Nets

Cited by 9 publications

References 33 publications

Torus bundles not distinguished by TQFT invariants

Torus bundles not distinguished by TQFT invariants

On Examples of Intermediate Subfactors from Conformal Field Theory

Congruence Subgroups and Generalized Frobenius-Schur Indicators

Contact Info

Product

Resources

About